# Neural Partial Differential Equations with Functional Convolution

arXiv: https://arxiv.org/abs/2303.07194 · HTML: https://mlsanity.com/paper/2303.07194/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 27 of its 40 references that are in the map.
- First posted on arXiv 10 Mar 2023.

## Builds on (most cited first, 27 of 27)

- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Relational inductive biases, deep learning, and graph networks](https://mlsanity.com/paper/1806.01261/) (2018)
- [DeepXDE: A deep learning library for solving differential equations](https://mlsanity.com/paper/1907.04502/) (2019)
- [Network In Network](https://mlsanity.com/paper/1312.4400/) (2013)
- [OptNet: Differentiable Optimization as a Layer in Neural Networks](https://mlsanity.com/paper/1703.00443/) (2017)
- [Solving high-dimensional partial differential equations using deep learning](https://mlsanity.com/paper/1707.02568/) (2017)
- [Interaction Networks for Learning about Objects, Relations and Physics](https://mlsanity.com/paper/1612.00222/) (2016)
- [Deep Convolutional Neural Network for Inverse Problems in Imaging](https://mlsanity.com/paper/1611.03679/) (2016)
- [Learning Deep CNN Denoiser Prior for Image Restoration](https://mlsanity.com/paper/1704.03264/) (2017)
- [Machine Learning for Fluid Mechanics](https://mlsanity.com/paper/1905.11075/) (2019)
- [Neural Relational Inference for Interacting Systems](https://mlsanity.com/paper/1802.04687/) (2018)
- [PDE-Net: Learning PDEs from Data](https://mlsanity.com/paper/1710.09668/) (2017)
- [PDE-Net 2.0: Learning PDEs from Data with A Numeric-Symbolic Hybrid Deep Network](https://mlsanity.com/paper/1812.04426/) (2018)
- [A Compositional Object-Based Approach to Learning Physical Dynamics](https://mlsanity.com/paper/1612.00341/) (2016)
- [Flexible Neural Representation for Physics Prediction](https://mlsanity.com/paper/1806.08047/) (2018)
- [Learning Neural PDE Solvers with Convergence Guarantees](https://mlsanity.com/paper/1906.01200/) (2019)
- [Learning Compositional Koopman Operators for Model-Based Control](https://mlsanity.com/paper/1910.08264/) (2019)
- [Propagation Networks for Model-Based Control Under Partial Observation](https://mlsanity.com/paper/1809.11169/) (2018)
- [Deep Learning Approximation for Stochastic Control Problems](https://mlsanity.com/paper/1611.07422/) (2016)
- [Forward-Backward Stochastic Neural Networks: Deep Learning of High-dimensional Partial Differential Equations](https://mlsanity.com/paper/1804.07010/) (2018)
- [Augmenting Physical Simulators with Stochastic Neural Networks: Case Study of Planar Pushing and Bouncing](https://mlsanity.com/paper/1808.03246/) (2018)
- [Enforcing Statistical Constraints in Generative Adversarial Networks for Modeling Chaotic Dynamical Systems](https://mlsanity.com/paper/1905.06841/) (2019)
- [Solving inverse-PDE problems with physics-aware neural networks](https://mlsanity.com/paper/2001.03608/) (2020)
- [Neural-net-induced Gaussian process regression for function approximation and PDE solution](https://mlsanity.com/paper/1806.11187/) (2018)
- [Learning to Discretize: Solving 1D Scalar Conservation Laws via Deep Reinforcement Learning](https://mlsanity.com/paper/1905.11079/) (2019)
- [FiniteNet: A Fully Convolutional LSTM Network Architecture for Time-Dependent Partial Differential Equations](https://mlsanity.com/paper/2002.03014/) (2020)
- [Coercing Machine Learning to Output Physically Accurate Results](https://mlsanity.com/paper/1910.09671/) (2019)

## Built on by (most cited first, 1 of 1)

- [Discovery and inversion of the viscoelastic wave equation in inhomogeneous media](https://mlsanity.com/paper/2409.18370/) (2024)

## Abstract (arXiv)

We present a lightweighted neural PDE representation to discover the hidden structure and predict the solution of different nonlinear PDEs. Our key idea is to leverage the prior of ``translational similarity'' of numerical PDE differential operators to drastically reduce the scale of learning model and training data. We implemented three central network components, including a neural functional convolution operator, a Picard forward iterative procedure, and an adjoint backward gradient calculator. Our novel paradigm fully leverages the multifaceted priors that stem from the sparse and smooth nature of the physical PDE solution manifold and the various mature numerical techniques such as adjoint solver, linearization, and iterative procedure to accelerate the computation. We demonstrate the efficacy of our method by robustly discovering the model and accurately predicting the solutions of various types of PDEs with small-scale networks and training sets. We highlight that all the PDE examples we showed were trained with up to 8 data samples and within 325 network parameters.